A Triadic Suffix Tokenization Scheme for Numerical Reasoning

Tokenization in language models is treated by default as an input preprocessing decision. We argue that this framing is incomplete: in autoregressive models, tokenizer granularity determines what the model must resolve in a single forward pass, and therefore the supervision signal it receives. This affects both the difficulty of the learning problem and the representations that emerge inside the model. We test this in a controlled experiment on numeric reasoning with a novel decoupling of input and output tokenization. As the output supervision view predicts, differences in task performance, training dynamics, and model internals are induced by output tokenization and largely invariant to input tokenization. This may matter in practice, because models with different tokenization strategies differ not only in input representation but in the task they were trained on. Comparisons between models may thus partly reflect task definition rather than ability. A survey of 120 recent *CL papers on numeric reasoning confirms that this is rarely acknowledged: only about 10% report the numeric tokenization of the models they evaluate, while 69% compare across tokenization, and thus supervision, regimes without reporting it. While prior work documents that tokenization consistently affects model performance, there is no principled account of why. We argue that framing tokenization as output supervision provides that account.

Supervised fine-tuning (SFT) applies a uniform cross-entropy loss to all target tokens, even though different tokens provide unequal learning signals for mathematical reasoning. This uniform treatment can over-sharpen already mastered tokens while amplifying learning pressure on uncertain, low-confidence tokens, leading to suboptimal training dynamics. We propose Trimmed Logit-Gap SFT (TrimSFT), a simple token-level reweighting method that scales the SFT loss according to the logit gap between the gold token and its strongest competitor. TrimSFT trims supervision away from both extremes: tokens already mastered (large logit gap) and tokens weakly supported by the current model (small or negative logit gap), concentrating learning within an intermediate logit-gap region between them. We instantiate this principle with a Gaussian weight centered at margin m with bandwidth τ, requiring no reference model or additional forward pass. We evaluate TrimSFT on six base models from the Llama, Qwen, and DeepMath families across five mathematical reasoning benchmarks. TrimSFT consistently improves over standard SFT, achieving the best average performance on five out of six models, with gains of up to +26.9 points over SFT on MATH500. Further analyses show that the bandwidth τ matters more than the exact margin location, and that half-trim variants that remove supervision pressure from only one side yield inferior trade-offs. A token-level logit-gap distribution analysis suggests that TrimSFT reshapes model confidence in a more balanced way than uniform SFT or monotonic reweighting methods. These results suggest that reasoning SFT can benefit from trimming both extremes rather than treating all tokens uniformly.

Tokenization is the first step in every language model (LM), yet it never takes the sounds of words into account. We investigate how tokenization influences text-only LMs' ability to represent phonological knowledge. Through a series of probing experiments, we show that subword-based tokenization systematically weakens the encoding of both local (e.g., rhyme) and global (e.g., syllabification) phonological features. To quantify this effect, we introduce the syllabification-tokenization alignment distance (STAD), a metric that measures the misalignment between a model's tokenization and the natural syllable boundaries of words, and find that higher misalignment correlates with poorer phonological representations, providing a simple diagnostic for phonology-aware tokenization. To address these limitations, we propose a lightweight IPA-based fine-tuning method that infuses phonological awareness into LMs, leading to consistent improvements across three phonology-related tasks while largely preserving math and general reasoning ability, with 1.1% and 0.9% drops on GSM8K and MMLU, respectively.